Abstract

In this paper, we introduce the concept of αc-admissible non-self mappings and prove the existence and convergence of the past-present-future (briefly, PPF) dependent fixed point theorems for such mappings in the Razumikhin class. We use these results to prove the PPF dependent fixed point of Bernfeld et al. (Appl. Anal. 6:271-280, 1977) and also apply our results to PPF dependent coincidence point theorems.

MSC:47H09, 47H10.

Keywords

PPF fixed pointsRazumikhin classesrational type contraction

1 Introduction

The applications of fixed point theory are very important and useful in diverse disciplines of mathematics. The theory can be applied to solve many problem in real world, for example: equilibrium problems, variational inequalities and optimization problems. A very powerful tool in fixed point theory is the Banach fixed point theorem or Banach’s contraction principle for a single-valued mapping. It is no surprise that there is a great number of generalizations of this principle. Several mathematicians have gone in several directions modifying Banach’s contractive condition, changing the space or extending a single-valued mapping to a multivalued mapping (see [1–10]).

One of the most interesting results is the extension of Banach’s contraction principle in case of non-self mappings. In 1997, Bernfeld et al. [11] introduced the concept of fixed point for mappings that have different domains and ranges, the so called past-present-future (briefly, PPF) dependent fixed point or the fixed point with PPF dependence. Furthermore, they gave the notion of Banach-type contraction for a non-self mapping and also proved the existence of PPF dependent fixed point theorems in the Razumikhin class for Banach-type contraction mappings. These results are useful for proving the solutions of nonlinear functional differential and integral equations which may depend upon the past history, present data and future consideration. Several PPF dependence fixed point theorems have been proved by many researchers (see [12–15]).

On the other hand, Samet et al. [16] were first to introduce the concept of α-admissible self-mappings and they proved the existence of fixed point results using contractive conditions involving an α-admissible mapping in complete metric spaces. They also gave some examples and applications to ordinary differential equations of the obtained results. Subsequently, there are a number of results proved for contraction mappings via the concept of α-admissible mapping in metric spaces and other spaces (see [17–19] and references therein).

To the best of our knowledge, there has been no discussion so far concerning the PPF dependent fixed point theorems via α-admissible mappings. In this paper, we introduce the concept of αc-admissible non-self mappings and establish the existence and convergence of PPF dependent fixed point theorems for contraction mappings involving αc-admissible non-self mappings in the Razumikhin class. Furthermore, we apply our results to the existence of PPF dependent fixed point theorems in [11] and also apply to PPF dependent coincidence point theorems.

2 Preliminaries

Throughout this paper, E denotes a Banach space with the norm ∥⋅∥E, I denotes a closed interval [a,b] in ℝ, and E0=C(I,E) denotes the set of all continuous E-valued functions on I equipped with the supremum norm ∥⋅∥E0 defined by

∥ϕ∥E0=supt∈I∥ϕ(t)∥E

for ϕ∈E0.

For a fixed element c∈I, the Razumikhin or minimal class of functions in E0 is defined by

Rc={ϕ∈E0:∥ϕ∥E0=∥ϕ(c)∥E}.

It is easy to see that the constant function is one of the mapping in Rc. The class Rc is said to be algebraically closed with respect to difference if ϕ−ξ∈Rc whenever ϕ,ξ∈Rc. Also, we say that the class Rc is topologically closed if it is closed with respect to the topology on E0 generated by the norm ∥⋅∥E0.

This implies that the sequence {ϕn} is a Cauchy sequence in Rc⊆E0. By the completeness of E0, we get that {ϕn} converges to a limit point ϕ∗∈E0, that is, limn→∞ϕn=ϕ∗. Since Rc is topologically closed, we have ϕ∗∈Rc.

Now we prove that ϕ∗ is a PPF dependent fixed point of T. By (d), we have α(ϕ∗(c),Tϕ∗)≥1. From assumption (c), we get

This implies that Tϕ∗=ϕ∗(c) and so ϕ∗ is a PPF dependent fixed point of T in Rc.

Finally, we prove the uniqueness of a PPF dependent fixed point of T in Rc. Let ϕ∗ and ξ∗ be two PPF dependent fixed points of T in Rc such that α(ϕ∗(c),Tϕ∗)≥1 and α(ξ∗(c),Tξ∗)≥1. By assumption (c), we have

and then ∥ϕ∗−ξ∗∥E0≤k∥ϕ∗−ξ∗∥E0. Since 0≤k<1, we get ∥ϕ∗−ξ∗∥E0=0 and then ϕ∗=ξ∗. Therefore, T has a unique PPF dependent fixed point in Rc. This completes the proof. □

Remark 3.6 If the Razumikhin class Rc is not topologically closed, then the limit of the sequence {ϕn} in Theorems 3.3, 3.4 and 3.5 may be outside of Rc, which may not be unique.

4 Consequences

In this section, we show that many existing results in the literature can be deduced from and applied easily to our theorems.

4.1 Banach contraction theorem

By applying Theorems 3.3, 3.4 and 3.5, we obtain the following results.

Theorem 4.1LetT:E0→E, and there exists a real numberk∈[0,1)such that

∥Tϕ−Tξ∥E≤k∥ϕ−ξ∥E0

(4.1)

for allϕ,ξ∈E0.

If there existsc∈Isuch thatRcis topologically closed and algebraically closed with respect to difference, thenThas a unique PPF dependent fixed point inRc.

Moreover, for a fixedϕ0∈Rc, if a sequence{ϕn}of iterates ofTinRcis defined by

Tϕn−1=ϕn(c)

(4.2)

for alln∈N, then{ϕn}converges to a PPF dependent fixed point ofTinRc.

Proof Let α:E×E→[0,∞) be the mapping defined by α(x,y)=1 for all x,y∈E. Then T is an αc-admissible mapping. It is easy to show that all the hypotheses of Theorems 3.3, 3.4 and 3.5 are satisfied. Consequently, T has a unique PPF dependent fixed point in Rc. □

4.2 PPF dependent coincidence point theorems

In this section, we discuss some relation between PPF dependent fixed point results and PPF dependent coincidence point results. First, we give the concept of PPF dependent coincidence point.

Definition 4.2 Let S:E0→E0 and T:E0→E. A point ϕ∈E0 is said to be a PPF dependent coincidence point or a coincidence point with PPF dependence of S and T if Tϕ=(Sϕ)(c) for some c∈I.

Definition 4.3 Let c∈I and S:E0→E0, T:E0→E, α:E×E→[0,∞). We say that (S,T) is an αc-admissible pair if for ϕ,ξ∈E0,

α((Sϕ)(c),(Sξ)(c))≥1impliesα(Tϕ,Tξ)≥1.

Remark 4.4 It easy to see that if (S,T) is an αc-admissible pair and S is an identity mapping, then T is also an αc-admissible mapping.

Now, we indicate that Theorem 3.3 can be utilized to derive a PPF dependent coincidence point theorem.

Theorem 4.5LetS:E0→E0, T:E0→E, α:E×E→[0,∞)be three mappings satisfying the following conditions:

(a)

There existsc∈Isuch thatS(Rc)is topologically closed and algebraically closed with respect to difference.

Proof Consider the mapping S:E0→E0. We obtain that there exists F0⊆E0 such that S(F0)=S(E0) and S|F0 is one-to-one. Since T(F0)⊆T(E0)⊆E, we can define a mapping A:S(F0)→E by

A(Sϕ)=Tϕ

(4.3)

for all ϕ∈F0. Since S|F0 is one-to-one, then A is well defined.

From (4.3) and condition (c), we have

α((Sϕ)(c),A(Sϕ))α((Sξ)(c),A(Sξ))∥A(Sϕ)−A(Sξ)∥E≤k∥Sϕ−Sξ∥E0

for all Sϕ,Sξ∈S(E0). This shows that A satisfies condition (c) of Theorem 3.3.

Now, we use Theorem 3.3 with a mapping A, then there exists a unique PPF dependent fixed point φ∈S(F0) of A, that is, Aφ=φ(c) and α(φ(c),Aφ)≥1. Since φ∈S(F0), we can find ω∈F0 such that φ=Sω. Therefore, we get

Tω=A(Sω)=Aφ=φ(c)=(Sω)(c)

and

α((Sω)(c),Tω)=α(φ(c),Aφ)≥1.

This implies that ω is a PPF dependent coincidence point of T and S. This completes the proof. □

Similarly, we can apply Theorems 3.4 and 3.5 to the Theorems 4.6 and 4.7. Then, in order to avoid repetition, the proof is omitted.

Theorem 4.6LetS:E0→E0, T:E0→E, α:E×E→[0,∞)be three mappings satisfying the following conditions:

(a)

There existsc∈Isuch thatS(Rc)is topologically closed and algebraically closed with respect to difference.

Declarations

Acknowledgements

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under NRU-CSEC project No. NRU56000508).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Authors’ Affiliations

(1)

Department of Mathematics, Texas A&M University-Kingsville

(2)

Department of Mathematics, Faculty of Science, King Abdulaziz University

(3)

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)

(4)

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center

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